Question 1:
Draw the graph of each of the following linear equations in two variables:
(i)x + y = 4
(ii)x – y = 2
(iii)y = 3x
(iv) 3 = 2x + y
Solution 1:
(i)x + y = 4
Given : Linear Equation
x + y = 4 ⇒ y = 4 – x ------------------Equation (1)
By substituting the different values of x in the Equation (1) we get different values for y
When x = 0 , we have : y = 4 – 0 = 4
When x = 2, we have : y = 4 – 2 = 2
When x = 4, we have : y = 4 – 4 = 0
Thus, we have the following table with all the obtained solutions:
(ii)x – y = 2
Given : Linear Equation
x - y = 2 ⇒ y = x – 2 ------------------Equation (1)
By substituting the different values of x in the Equation (1) we get different values for y
When x = 0 , we have y = 0 – 2 = – 2
When x = 2, we have y = 2 – 2 = 0
When x = 4, we have y = 4 – 2 = 2
Thus, we have the following table with all the obtained solutions:
(iii)y = 3x
Given: Linear Equation
y = 3x ------------------Equation (1)
By substituting the different values of x in the Equation (1) we get different values for y
When x = 0 , we have : y = 3(0) = 0
When x = 1, we have : y = 3 (1) = 3
When x = –1, we have : y = 3(–1) = – 3
Thus, we have the following table with all the obtained solutions:
(iv)3 = 2x + y
Given: 3 = 2x + y be the Linear Equation
⇒y = 3 – 2x ------------------Equation (1)
By substituting the different values of x in the Equation (1) we get different
values for y
When x = 0, we have: y = 3 – 2(0) = 3 – 0 = 3
When x = 3, we have: y = 3 – 2(3) = 3 – 6 = – 3
When x = – 1, we have: y = 3 – 2 (–1) = 3 + 2 = 5
Thus, we have the following table with all the obtained solutions:
The graph of the line represented by the given equation as shown.
Question 2:
Given the equations of two lines passing through (2, 14). How many more such lines are there, and why?
Solution 2:
Given :
Equations of two lines passing through (2, 14).
It can be observed that point (2, 14) satisfies the equation 7x − y = 0 and x − y + 12 = 0.
Therefore, 7x − y = 0 and x − y + 12 = 0 are two lines passing through point (2, 14).
As it is known that through one point, infinite number of lines can pass through, therefore, there are infinite lines of such type passing through the given point.
Question 3:
If the point (3, 4) lies on the graph of the equation 3y = ax + 7, find the value of a.
Solution 3:
Given :
3y = ax + 7 is the Linear Equation -----------Equation(1)
Point (3, 4) lies on the Equation (1)
By Substituting the value of x = 3 and y = 4 in the Equation (1),
3y = ax + 7
3(4) = a(3) + 7
5 = 3a
We get,
a=5/3
Question 4:
The taxi fare in a city is as follows: For the first kilometre, the fares is Rs. 8 and for the subsequent distance it is Rs. 5 per km. Taking the distance covered as x km and total fare as Rs. y, write a linear equation for this information, and draw its graph.
Solution 4:
Given,
Fare for 1st kilometer = Rs. 8
Taxi fare for the subsequent km = Rs 5
Fare for the rest of the distance = Rs. (x − 1) 5
Let the Total distance covered = x km
Let the Total fare covered = Rs. y
The linear equation for the above information is given by,
Total fare, y = [8 + (x − 1) 5]
y = 8 + 5x − 5
y = 5x + 3
5x − y + 3 = 0
By substituting the different values of x in the Equation (1) we get different values for y
When x = 0, y = 5 × 0 + 3 = 0 + 3 = 3
When x = 1, y = 5 ×( 1) + 3 = 5 + 3 = 8
When x = 2, y = 5 × (2) + 3 = 10 + 3 = 13
When x = -1, y = 5 × (–1) + 3 = –5 + 3 = -2
When x = -2, y = 5 × (–2) + 3 = –10 + 3 = -7
Thus, we have the following table with all the obtained solutions:
Here, it can be seen that variable x and y are representing the distance covered and the fare paid for that distance respectively and these quantities may not be negative.
Hence, only those values of x and y which are lying in the 1st quadrant will be considered.
Question 5:
From the choices given below, choose the equation whose graphs are given in the given figures.
For the first figure For the second figure
(i)y = x (i)y = x +2
(ii)x + y = 0 (ii)y = x − 2
(iii)y = 2x (iii)y = − x + 2
(iv)2 + 3y = 7x (iv)x + 2y = 6
Solution 5:
For the First Figure:
Points on the given line are (−1, 1), (0, 0), and (1, −1).
It can be observed that the coordinates of the points of the graph satisfy
the equation x + y = 0.
Therefore, x + y = 0 is the equation corresponding to the graph as shown in the first figure.
Hence, (ii) is the correct answer.
For the Second Figure :
Points on the given line are (−1, 3), (0, 2), and (2, 0).
It can be observed that the coordinates of the points of the graph satisfy the equation y = −x + 2.
Therefore, y = −x + 2 is the equation corresponding to the graph shown in the second figure.
Hence, (iii) is the correct answer.
Question 6:
If the work done by a body on application of a constant force is directly proportional to the distance travelled by the body, express this in the form of an equation in two variables and draw the graph of the same by taking the constant force as 5 units. Also read from the graph the work done when the distance travelled by the body is
(i)2 units (ii)0 units
Solution 6:
Given:
Let the distance travelled and the work done by the body be x and y respectively.
Work done ∝ distance travelled
Hence, y ∝ x
y = kx ----------- Equation (1)
Where, k is a constant
By substituting the different values of x in the Equation (1) we get different
values for y
When x= 0, y = 0
When x= 1, y = 5
When x =-1, y= -5
Thus, we have the following table with all the obtained solutions:
(i)From the graphs, it can be observed that the value of y corresponding to x = 2 is 10. This implies that the work done by the body is 10 units when the distance travelled by it is 2 units.
(ii)From the graphs, it can be observed that the value of y corresponding to x = 0 is 0. This implies that the work done by the body is 0 units when the distance travelled by it is 0 unit.
Question 7:
Yamini and Fatima, two students of Class IX of a school, together contributed Rs. 100 towards the Prime Minister’s Relief Fund to help the earthquake victims. Write a linear equation which satisfies this data. (You may take their contributions as Rs. x and Rs. y.) Draw the graph of the same.
Solution 7:
Given,
Let the amount that Yamini and Fatima contributed be x and y respectively towards the Prime Minister’s Relief fund.
Amount contributed by Yamini and Fatima together would be = 100
x + y = 100
y = 100- x
By substituting the different values of x in the Equation (1) we get different values for y
When x = 0, y = 100
When x = 50, y = 50
When x = 100, y = 0
Thus, we have the following table with all the obtained solutions:
Here, it can be seen that variable x and y are representing the amount contributed by Yamini and Fatima respectively and these quantities cannot be negative. Hence, only those values of x and y which are lying in the 1st quadrant will be considered.
Question 8:
In countries like USA and Canada, temperature is measured in Fahrenheit, whereas in countries like India, it is measured in Celsius. Here is a linear equation that converts Fahrenheit to Celsius:
F=(9/5)C+32
(i)Draw the graph of the linear equation above using Celsius for x-axis and Fahrenheit for y-axis.
(ii)If the temperature is 30°C, what is the temperature in Fahrenheit?
(iii)If the temperature is 95°F, what is the temperature in Celsius?
(iv)If the temperature is 0°C, what is the temperature in Fahrenheit and if the temperature is 0°F, what is thetemperature in Celsius?
(v)Is there a temperature which is numerically the same in both Fahrenheit and Celsius? If yes, find it.
Solution 8:
(i) Given,
F=(9/5)C+32-------------------Equation(1)
By substituting the different values of x in the Equation (1) we get different values for y
When C = 0,
F=(9/5)0+32
When C = – 40, F = -40
When C = 10, F = 50
Thus, we have the following table with all the obtained solutions:
